Reduced-order modelling of unsteady cavitating flow around a Clark-Y hydrofoil

This paper proposes a novel approach that combines Proper Orthogonal Decomposition (POD) reduced-order system with Long Short-Term Memory (LSTM) neural network to predict flow velocity. Large Eddy Simulation (LES) is used to simulate the cavitating flow around a NACA66 hydrofoil. POD is adopted to reduce the dimensionality of the high-dimensional data. It was found that 66.81% of the flow field energy and dominant coherent structures can be captured with first eight POD modes. The LSTM network model was further used to predict the temporal data of the POD mode coefficients, and the error of the predicted coefficients was within an acceptable range. The reconstructed flow field agrees well with the real flow field and the cavitation development has also been well illustrated. This method provides a promising and efficient alternative for flow prediction and has potential for applications in fluid dynamics, aerospace engineering, and hydrodynamics.


Introduction.
Cavitation occurs when liquid pressure falls below vapor pressure, causing issues like vibration [1,2] and noise in hydraulic machinery.Understanding these dynamics is vital for hydraulic system reliability.Scholars have explored cavitation through both empirical and numerical methods [3,4].Numerical techniques, including DNS, RANS, LES, and Hybrid Models, are employed to analyze turbulence.
Unsteady cavitating flows are challenging to analyze due to high-dimensional data.Reduced Order Models (ROMs), like POD [5], have been used to reduce costs.Advanced machine learning techniques like LSTM have been introduced in fluid mechanics.Deep learning's success in various fields led to its application in fluid mechanics [6].
Hydrodynamic characteristics around hydrofoils impact efficiency and cavitation performance.Integrating POD [7] and LSTM [8] techniques for cavitation studies in hydrofoils is unexplored.This paper introduces a novel framework harmonizing POD and LSTM for predicting velocity near a hydrofoil.This approach offers efficient temporal prediction and spatial reconstruction, enhancing higher-fidelity modeling.
The paper is structured as follows: Section 2 outlines governing equations and the LES approach.In Section 3, we present the integrated framework, followed by computational results in Section 4. The paper concludes in Section 5 summarizing findings.

LES approach.
The current study employs the Large Eddy Simulation (LES) technique, coupled with the homogenous cavitation model, to delve into the hydrodynamic intricacies.A comprehensive investigation of the unsteady cavitation flow around a Clark-Y hydrofoil has been conducted through a three-dimensional numerical simulation.The hydrofoil possesses a chord length denoted as C, which is 70 mm, and is set at an angle of attack of 8 degrees.The computational domain and configuration are depicted in Figure 1.The computational section's length and height are specified as 10C and 2.7C, respectively.To mitigate the substantial computational demands that would arise from adopting the actual spanwise dimension of the experimental setup, a symmetric assumption was imposed on the two lateral boundaries.The inlet boundary condition is prescribed with a normal velocity of U∞ = 10 m/s, while the static pressure at the outlet was set to 43468 Pa.This approach is adopted to meticulously emulate the experimental conditions, thereby enabling a perceptive juxtaposition of the numerical and experimental findings.The upper and lower walls are subjected to free-slip wall conditions, ensuring minimal frictional effects.In contrast, the surface of the hydrofoil is treated as a non-slip wall, capturing the interactions with the fluid more accurately.The lateral boundary conditions are imposed as periodic conditions, ensuring continuity and minimizing artificial boundary effects.The time step was set to 2.8* 10 -5 s.More details, can be referred to Ref [9,10].

ROM -LSTM methodology.
The primary objective of this research is to construct an LSTM-POD Neural Network System (NNS) tailored for forecasting modal coefficients.The LSTM-ROM procedure can be succinctly summarized as follows: In this study, a systematic methodology is employed.The process begins with dataset selection, wherein snapshot datasets obtained from LES simulations are carefully curated to form a set of M planes, selected to serve as both training and test datasets.Subsequently, the computation of dominant modes and the corresponding time-dependent modal coefficients an m is carried out using advanced Proper Orthogonal Decomposition

Results.
In this section, a comparative analysis was conducted between the experimental results obtained using Particle Image Velocimetry technology [10] and those from numerical simulation.Figure 3 illustrates a sequential series of images capturing the shedding of the cavity within a representative cycle of period T. Huang's investigation suggested that the re-entrant jet is identified as the primary mechanism accountable for the cavity shedding phenomenon in the context of the Clark-Y hydrofoil.Following this, a fresh cycle initiate.Notably, the progression of unsteady cavitation in the numerical simulation concurs remarkably with the observed experimental pattern.Upon closer examination, it's evident that the predicted values by the a1 and a2 models closely approximate the actual values.However, as the order of magnitude increases, the prediction accuracy diminishes.This decline can be attributed to the higher complexity of training data associated with LSTM models containing larger modal coefficients.Such datasets exhibit intricate patterns and challenging-to-predict peaks.From a physical perspective, higher-order POD modes encompass more conspicuous behaviors like cavity detachment and the re-entrant jet in cloud cavitation.Conversely, lower-order POD modes correlate with smaller-scale vortices marked by high frequencies, representing nonlinear characteristics.To comprehensively explore the POD-LSTM model's predictive prowess, an in-depth analysis focused on reconstructed velocities at six strategically chosen hydrofoil points across 400 prediction time steps.
These results are visually depicted in Figure 5 and succinctly presented in Figure 6.These figures unequivocally showcase the model's remarkable capability in accurately capturing peak velocities and their corresponding temporal occurrences.Of particular interest is the distinct trend observed in the error profiles.Notably, the POD and POD-LSTM models exhibit a remarkable reduction in errors compared to the discrepancies witnessed between the LES and POD-LSTM method.This intriguing contrast in error characteristics can be attributed to the inherent attributes of the POD methodology.
By acting as an effective filter, the POD approach adeptly sieves out negligible minutiae, while retaining the essence of vital information.This key attribute significantly contributes to the enhanced predictive performance demonstrated by the POD-based models.This approach that synergistically harnesses the strengths of both the POD and LSTM methodologies.This synergistic fusion not only enriches our understanding of intricate unsteady flow phenomena but also augments their accurate prediction.Ultimately, the dynamic interplay between the model's ability to decipher essential flow features and its predictive prowess heralds a new era of comprehensive comprehension and accurate forecasting of the complex fluid dynamics around hydrofoils.The velocity prediction results of the six supervised points.Figure 7 illustrates reconstructed velocity flow fields at six discrete time intervals, along with error analyses utilizing both POD and POD-LSTM techniques.The outcomes of this analysis provide encouraging insights, particularly regarding prediction quality.These selected time intervals effectively demonstrate the evolution of unsteady cavitation dynamics, highlighting the model's capability in capturing the complexities of this phenomenon.To comprehensively assess errors, the Mean Absolute Percentage Error (MAPE) metric is employed as a quantitative measure of predictive precision.The formula for Error is given by:

MAPE  actual(t)-forecast(t) actual(t)
Where n represents the number of samples, actual(t) denotes the actual value, and forecast(t) signifies the predicted value at each time step.
The notable alignment between the predicted and actual values during these pivotal time steps underscores the efficacy of the POD-LSTM framework in capturing the intricate dynamics of unsteady cavitation around the hydrofoil.

Conclusion.
In conclusion, this study has harnessed the synergistic potential of Proper Orthogonal Decomposition (POD) and Long Short-Term Memory (LSTM) neural networks to enhance our understanding of unsteady cavitation dynamics around hydrofoils.By efficiently extracting dominant flow modes using POD and subsequently utilizing LSTM for predictive modeling, we have effectively bridged the gap between reduced-order modeling and deep learning methodologies.The proposed LSTM-POD framework has demonstrated remarkable promise in accurately predicting the temporal evolution and spatial characteristics of cloud cavitation.This approach not only offers an innovative paradigm for studying complex fluid phenomena but also showcases its applicability in real-world engineering scenarios.
Moving forward, several avenues for further exploration emerge from this work.First, extending the current methodology to more intricate geometries and diverse flow conditions would undoubtedly enrich our insights.Moreover, exploring the potential integration of advanced machine learning techniques like transformers could potentially enhance the model's predictive capabilities and temporal extrapolation.Additionally, investigating the transferability of the proposed approach to other fluid dynamic phenomena and the incorporation of uncertainty quantification methods could expand its utility and robustness.
In summary, this study's success in leveraging the synergy between POD and LSTM underscores the growing importance of multidisciplinary approaches in tackling complex fluid dynamics challenges.As technology continues to advance and our understanding deepens, the prospects for
(POD) techniques, constituting the Dominant Mode and Coefficient Computation phase.Through the application of POD techniques for modal energy computation, we determined that the initial eight modes encompass 66.81% of the flow field's energy.As a result, our study exclusively employed these first eight modes for coefficient prediction and modeling.Then the Training LSTM Neural Networks stage involves the construction of intricate LSTM neural network architectures.This construction involves iterative input-output interactions, encapsulating timedependent modal temporal coefficients.The LSTM model's training data comprises time-dependent modal coefficients derived from identical modes across different planes.The construction of training input dataset P for the LSTM model involves {aP (tn), aP (tn +1), …, aP (tm), … aP (tn +T-1)} as input, while the corresponding output dataset constitutes { aP (tn +T), aP (tn +T+1), …, aP (tn +T+m), … aP (tn +2T-1)}.Following training, the Prediction and Comparison phase employs the trained LSTM model M to predict test data a(t + t)' at the subsequent instance of each mode.A comprehensive analysis is conducted to analyze the predictive accuracy by comparing the predicted values with the actual counterparts a(t + t)', thereby quantifying uncertainties.Finally, the methodology also encompasses Velocity Field Reconstruction, utilizing LSTM model-predicted temporal coefficients to accurately reconstruct the intricate velocity field.

Figure 2 .
Figure 2. The framework of the LSTM-ROM methodology.

Figure 3 .
Figure 3.Comparison between experimental patterns[10] and the predicted side views of the surface, αv = 0.2.σ = 0.8, α= 8 °, U∞ = 10 m/s.To comprehensively evaluate the trained LSTM model's effectiveness, it was employed on the test dataset for assessment.The assessment results are depicted in Figure4, showcasing the comparison between predicted and actual values from the testing dataset.Notably, the input data consists of 100-time steps, and the model is iteratively used for generating predictions.Upon closer examination, it's evident that the predicted values by the a1 and a2 models closely approximate the actual values.However, as the order of magnitude increases, the prediction accuracy diminishes.This decline can be attributed to the higher complexity of training data associated with LSTM models containing larger modal coefficients.Such datasets exhibit intricate patterns and challenging-to-predict peaks.From a physical perspective, higher-order POD modes encompass more conspicuous behaviors like cavity detachment and the re-entrant jet in cloud cavitation.Conversely, lower-order POD modes correlate with smaller-scale vortices marked by high frequencies, representing nonlinear characteristics.

Figure 4 .
Figure 4.The prediction results of POD coefficients.To comprehensively explore the POD-LSTM model's predictive prowess, an in-depth analysis focused on reconstructed velocities at six strategically chosen hydrofoil points across 400 prediction time steps.These results are visually depicted in Figure5and succinctly presented in Figure6.These figures unequivocally showcase the model's remarkable capability in accurately capturing peak velocities and their corresponding temporal occurrences.Of particular interest is the distinct trend observed in the error profiles.Notably, the POD and POD-LSTM models exhibit a remarkable reduction in errors compared to the discrepancies witnessed between the LES and POD-LSTM method.This intriguing contrast in error characteristics can be attributed to the inherent attributes of the POD methodology.By acting as an effective filter, the POD approach adeptly sieves out negligible minutiae, while retaining the essence of vital information.This key attribute significantly contributes to the enhanced predictive performance demonstrated by the POD-based models.This approach that synergistically harnesses the strengths of both the POD and LSTM methodologies.This synergistic fusion not only enriches our understanding of intricate unsteady flow phenomena but also augments their accurate prediction.Ultimately, the dynamic interplay between the model's ability to decipher essential flow features and its predictive prowess heralds a new era of comprehensive comprehension and accurate forecasting of the complex fluid dynamics around hydrofoils.

Figure 5 .Figure 6 .
Figure 5.The position of the supervised point on the hydrofoil.